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Mathematical Expression Editor

We study constant coefficient nonhomogeneous systems, making use of variation of
parameters to find a particular solution.

Variation of Parameters for Nonhomogeneous Linear Systems

We now consider the nonhomogeneous linear system
where is an matrix function and is an -vector forcing function. Associated with
this system is the complementary system .

The next theorem is analogous to Theorems thmtype:5.3.2 and thmtype:9.1.5. It shows how to find the
general solution of if we know a particular solution of and a fundamental
set of solutions of the complementary system. We leave the proof to the
reader.

Suppose the matrix function and the -vector function are continuous on . Let be a
particular solution of on , and let be a fundamental set of solutions of
the complementary equation on . Then is a solution of on if and only
if
where are constants.

Finding a Particular Solution of a Nonhomogeneous System

We now discuss an extension of the method of variation of parameters to linear
nonhomogeneous systems. This method will produce a particular solution of a
nonhomogenous system provided that we know a fundamental matrix for the
complementary system. To derive the method, suppose is a fundamental matrix for
the complementary system; that is,
where
is a fundamental set of solutions of the complementary system. In Trench 10.3 we
saw that . We seek a particular solution of

of the form
where is to be determined. Differentiating (eq:10.7.2) yields

Comparing this with (eq:10.7.1) shows that is a solution of (eq:10.7.1) if and only if
Thus, we can find a particular solution by solving this equation for , integrating to
obtain , and computing . We can take all constants of integration to be zero, since
any particular solution will suffice.

This method is analogous to the method of variation of parameters discussed in
Trench 5.7 and 9.4 for scalar linear equations.

(a)

Find a particular solution of the system
which we considered in Example example:10.2.1.

item:10.7.1a The complementary system is
The characteristic polynomial of the coefficient matrix is
Using the method of Trench 10.4, we find that
are linearly independent solutions of (eq:10.7.4). Therefore
is a fundamental matrix for (eq:10.7.4). We seek a particular solution of (eq:10.7.3), where ; that
is,
The determinant of is the Wronskian
By Cramer’s rule,
Therefore
Integrating and taking the constants of integration to be zero yields
so
is a particular solution of (eq:10.7.3).

which can also be written as
where is an arbitrary constant vector.

Writing (eq:10.7.5) in terms of coordinates yields

so our result is consistent with Example example:10.2.1.

If isn’t a constant matrix, it’s usually difficult to find a fundamental set of
solutions for the system . It is beyond the scope of this text to discuss methods
for doing this. Therefore, in the following examples and in the exercises
involving systems with variable coefficient matrices we’ll provide fundamental
matrices for the complementary systems without explaining how they were
obtained.

Find a particular solution of
given that
is a fundamental matrix for the complementary system.

We seek a particular solution of (eq:10.7.6) where ; that is,
The determinant of is the Wronskian
By Cramer’s rule,
Therefore
Integrating and taking the constants of integration to be zero yields
so
is a particular solution of (eq:10.7.6).

Find a particular solution of
given that
is a fundamental matrix for the complementary system on and .

We seek a particular solution of (eq:10.7.7) where ; that is,
The determinant of is the Wronskian
By Cramer’s rule,
Therefore
Integrating and taking the constants of integration to be zero yields
so
is a particular solution of (eq:10.7.7).

item:10.7.4a The complementary system for (eq:10.7.8) is
The characteristic polynomial of the coefficient matrix is
Using the method of Trench 10.4, we find that
are linearly independent solutions of (eq:10.7.9). Therefore
is a fundamental matrix for (eq:10.7.9). We seek a particular solution of (eq:10.7.8), where ; that
is,
The determinant of is the Wronskian
Thus, by Cramer’s rule,
Therefore
Integrating and taking the constants of integration to be zero yields
so
is a particular solution of (eq:10.7.8).

item:10.7.4b From Theorem thmtype:10.7.1 the general solution of (eq:10.7.8) is
which can be written as
where is an arbitrary constant vector.

Find a particular solution of
given that
is a fundamental matrix for the complementary system.

We seek a particular solution of (eq:10.7.10) in the form , where ; that is,
The determinant of is the Wronskian
By Cramer’s rule,
Therefore
Integrating and taking the constants of integration to be zero yields
so
is a particular solution of (eq:10.7.10).

Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored
and Edited Books & CDs. 8. (CC-BY-NC-SA)